did you test your conjecture for small values of n? for a 2x2 matrix:
which appears to have eigenvalues 0 and 2.
first he subtracted the first row, from every other row. this changed all the 1's in the first column (except for the top entry) to -λ.
it also changed all the 1's in every other column to 0, except along the diagonal, where 1-λ becomes -λ.
then he sucessively added (it doesn't matter if you do it one at a time, or all at once) columns 2 through n, to the first column.
neither the row operation, nor the column operation changes the determinant, since no row or column was multiplied.